The obvious assumption is that the coat of arms relates to the woman. If she is married they will be her husband’s arms, if unmarried, her father’s.
The arms in the first and fourth quarters of the shield were used by members of the Gyll family of county Durham and Yorkshire (according to Papworth’s Ordinary of British Arms) and by members of the Gillow or Gillon family in Kent (according to monumental inscriptions in the church of St Laurence, Thanet).
There are several possibilities for the arms in the second and third quarters, but because the colours are not shown it is not possible to distinguish between them – they were used by members of the Womvill, Preston, Shelley, Freeling, and Paris families.

As others have already guessed this is an engraving of Sarah Gilly, which matches up with the identification from the arms (Gyll/Gillow/Gillon). She is usually referred to as Mrs Sarah Gilly, though the description from the National Portrait Gallery identifies her as the daughter of Matthew Gilly. This minor mystery is easily explained: the title “Mrs” was a marker of social not marital status in this period. (See Amy Erickson’s article “Mistresses and Marriage: or, a Short History of the Mrs”, https://academic.oup.com/hwj/article-abstract/78/1/39/627183 )

However (and this is where I think the crocodile comes in) this same picture was used in the early works of Hannah Woolley, and was thought to be her, though by at least the late 18th century the correct identification was known or suspected.

I can’t say I’ve ever seen her before, but according to LUNA, the image is the frontispiece portrait of “A Guide to Ladies, Gentlewomen and Maids” (W3278.5), so my guess would be that this must be the book’s author, Hannah Woolley.

OK, after having looked up that portrait, I suppose the real answer might be something like “an old portrait of Sarah Gilly passed off as a portrait of Hannah Woolley by a cunning engraver, who had no idea what Mrs Woolley really looked like because he had never met her, and correctly assumed that readers would know neither face, so they wouldn’t notice”.